$$$\sin{\left(6 c \right)}$$$ 的积分

求$$$\int \sin{\left(6 c \right)}\, dc$$$。

设$$$u=6 c$$$。

则$$$du=\left(6 c\right)^{\prime }dc = 6 dc$$$ (步骤见»)，并有$$$dc = \frac{du}{6}$$$。

因此，

$${\color{red}{\int{\sin{\left(6 c \right)} d c}}} = {\color{red}{\int{\frac{\sin{\left(u \right)}}{6} d u}}}$$

对 $$$c=\frac{1}{6}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{\frac{\sin{\left(u \right)}}{6} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{6}\right)}}$$

正弦函数的积分为 $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{6} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{6}$$

回忆一下 $$$u=6 c$$$:

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{6} = - \frac{\cos{\left({\color{red}{\left(6 c\right)}} \right)}}{6}$$

因此，

$$\int{\sin{\left(6 c \right)} d c} = - \frac{\cos{\left(6 c \right)}}{6}$$

加上积分常数：

$$\int{\sin{\left(6 c \right)} d c} = - \frac{\cos{\left(6 c \right)}}{6}+C$$

$$$\int \sin{\left(6 c \right)}\, dc = - \frac{\cos{\left(6 c \right)}}{6} + C$$$A